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Explanation of the Gibbs Paradox within the Framework of Quantum Thermodynamics. Theo M. Nieuwenhuizen . Physikalisches Kolloquium Johann Wolfgang Goete Universitaet Frankfurt am Main 31-01, 2007. Outline.
Theo M. Nieuwenhuizen
Physikalisches KolloquiumJohann Wolfgang Goete Universitaet
Frankfurt am Main
Who was Josiah Willard Gibbs?What is the Gibbs Paradox?On previous explanations: mixing entropy
Crash course in Quantum Thermodynamics
Maximal work = ergotropy
Application of mixing ergotropy to the paradox
1839 – 1903
Carreer in Yale
1866-69: Travel to Paris, Berlin, Heidelberg
Gustav Kirchhoff, Hermann von Helmholtz
Gibbs free energy
Gibbs entropyGibbs ensemblesGibbs Duhem relation
Copley Medal 1901
But if A and B identical, no increase.
The paradox: There is a discontinuity, still k ln 2 for very similar but non-identical gases.
= N log 2
The paradox is solved within information theoretic approach to classical thermodynamics
Solution has been achieved within quantum statistical physics due to feature of partial distinguishability
Quantum physics is right starting point.But due to non-commutivity, the paradox is still unexplained.
Von Neuman entropy
ranges continuously from 2N ln 2 (orthogonal) to 0 (identical) .Many scholars believe this solves the paradox.
Dieks & van Dijk ’88: thermodynamic inconsistency, because there is no way to close the cycle by unmixing.If nonorthogonal to any attempt to unmix (measurement) will alter the states.
No thermodynamic limit
Bath has to be described explicitlyNon-negligible interaction energy
Langevin equation (if initially no correlation between S and B)
where H is that part of the total Hamiltonian,
that governs the unitary part of (Langevin) dynamicsin the small Hilbert space of the system.
Work: Energy-without-entropy added to the system bya macroscopic source.
1) Just energy increase of work source2) Gibbs-Planck: energy of macroscopic degree of freedom.
Energy related to uncontrollable degrees of freedom
Picture developed by Allahverdyan, Balian, Nieuwenhuizen ’00 -’04
B phase =Balian –Werthamer phase
- Eigenfrequencies of Schroedinger operators in finite domain
- Casimir effect: Balian-Duplantier sum rule
- Book: From microphysics to macrophysics- Quantum measurement process
No thermodynamic limit Thermodynamics endangered
Different formulations are inequivalent
-Generalized Thomson formulation is valid:
Cyclic changes on system in Gibbs equilibrium cannot yield work
(Pusz+Woronowicz ’78, Lenard’78, A+N ’02.)
- Rate of energy dispersion may be negative Classically: = T* ( rate of entropy production ): non-negative
A+N: PRL 00 ; PRE 02, PRB 02, J. Phys A 02
Experiments proposed for mesoscopic circuits and quantum optics.
statistical mechanicsquantum thermodynamicsquantum measurement process
astrophysics, cosmology, arrow of time
adiabatic theoremsquantum optics
quantum work fluctuations
> 35 common papers
Couple to work source and do all possible work extractions
Thermodynamics: minimize final energy at fixed entropyAssume final state is gibbsian: fix final T from S = const.Extracted work W = U(0)-U(final)
But: Quantum mechanics is unitary,
So all n eigenvalues conserved: n-1 constraints, not 1.
(Gibbs state typically unattainable for n>2)
Optimal final situation: eigenvectors of become those of H
Lowest final energy:highest occupation in ground state,one-but-highest in first excited state, etc(ordering )
(divine action, Aristotle)
Allahverdyan, Balian, Nieuwenhuizen, EPL 03.
-Comparison of activities:
Thermodynamic upper bounds: more work possible from
But actual work may be largest from
- Coupling to an auxiliary system : if is less active than
Then can be more active than
-Thermodynamic regime reduced to states that majorize one another
- Optimal unitary transformations U(t) do yield, in examples, explicit Hamiltonians for achieving optimal work extraction
L+N book: Thermodynamics of the glassy state
Gibbs paradox not solved up to nowMixing entropy argument has its own drawbacks
Explanation by formulation in terms of workMixing ergotropy = loss of maximal extractable work due to mixing
Operational definition: less work from less good apparatus
More mixing does not imply more work and vice versaMany details in Allahverdyan + N, Phys. Rev. E 73, 056120 (2006)
One of the formulations of the second law:
Adiabatic thermally isolated processes done on an equilibrium system are optimal (cost least work or yield most work)
In finite Q-systems: Work larger or equal to free energy difference
But adiabatic work is not free energy difference.
A+N, PRE 2003:
-No level crossing : adiabatic theorem holds
-Level crossing: solve using adiabatic perturbation theory.
Diabatic processes are less costly than adiabatic.Work = new tool to test level crossing.
Level crossing possible if two or more parameters are changed. Review expts on level crossing: Yarkony, Rev Mod Phys 1996